Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Abstract

Abstract. Quantum stochastic differential equations of the form dkt = kt • θ α β dΛ β α (t) govern stochastic flows on a C * -algebra A. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on A. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when A is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and * -homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations:, in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions. IntroductionThe quantum stochastic differential equation (QSDE)

A stochastic Stinespring Theorem

“…( [Par], [LW1]). To achieve this continuous time stochastic Stinespring decomposition it may be necessary to enlarge the dimension of the quantum noise driving the given process k. Interest in completely positive stochastic flows and their structure comes from several sources.…”

“…To achieve this continuous time stochastic Stinespring decomposition it may be necessary to enlarge the dimension of the quantum noise driving the given process k. Interest in completely positive stochastic flows and their structure comes from several sources. They arise naturally in the quantum theory of filtering ([Be1]); they will be required for any quantum theory of measure-valued diffusions ( [GLSW]); but also their study allows quantum dynamical semigroups ( [EvL]), quantum diffusions on an operator algebra ( [Hud], [Eva]), and contraction processes on a Hilbert space ( [Fag], [Moh]), to be viewed from a common vantage point ( [LiP], [LW1]). In a sister paper ( [GLSW]) completely positive Markovian contraction cocycles are dilated to * -homomorphic Markovian cocycles driven by a higher dimensional quantum noise, extending the dilation of quantum dynamical semigroups achieved in [GoS].…”

Quantum stochastic semigroups and their generators

“…Let us prove that the germ-maps γ µ ν of a CP flow φ must be conditionally completely positive (CCP) in a degenerated sense as it was found for the finite-dimensional bounded case in [7,11]. Another, equivalent, but not so explicit characterization was suggested for this particular case in [12]. The proof is given in [9,10] even for the general (noncommutative) algebras a and A.…”

“…As was proven in [7,11] for the case of finite-dimensional matrix γ of bounded γ µ ν , see also [12], the matrix elements K − ν can be chosen in such way that the matrix map ϕ = (ϕ µ ν )…”

“…The existence of minimal CP solution which has been recently constructed under certain continuity conditions in [10] proves that this structure is also sufficient for the CP property of any solution to this stochastic equation. The construction of the differential dilations and the CP solutions of such quantum stochastic differential equations with the bounded generators over the simple finitedimensional Itô algebra a = Q (E) and the arbitrary B ⊆ L (H) was also recently discussed in [11,12].…”